# ✪✪✪ Metamagical Themas Analysis

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Digital images using only two colors are called binary images or boolean images. Figure 2: The matrix corresponding to Felix The Cat. Grayscale images can also be represented by matrices. Each element of the matrix determines the intensity of the corresponding pixel. For convenience, most of the current digital files use integer numbers between to indicate black, the color of minimal intensity and to indicate white, maximum intensity , giving a total of different levels of gray This quantity of levels of gray is reasonable to work with images in WEB pages. However, there are certain specific applications that need more levels of gray in order to reproduce the image with more details and avoid rounding errors in numerical calculations, as is the case of medical images.

Color images , in turn, can be represented by three matrices. Each matrix specifies the amount of red , green and blue that makes up the image. The elements of these matrices are integer numbers between and , and they determine the intensity of the pixel with respect to the color of the matrix. Thus, in the RGB system, it is possible to represent different colors. Figure 3: Original picture, Red, Green and Blue components.

Once a digital image can be represented by matrices, we may ask how operations on their elements affect the corresponding image. For example, if we consider the binary image below as a matrix, say , then the image corresponds to the transposed matrix of , that is,. The image , by its turn, corresponds to the matrix. Try to discover the matricial relationships between the image and the other images! Another example: if we take the standard arithmetic mean of the component matrices , and from a color image , we will get a grayscale version of the image non-integer values are rounded to the nearest integer :.

Figure 5: arithmetic mean of the component matrices. One more example: using the operations of multiplication by a scalar and sum of matrices, it is possible to create an image transition effect commonly used, for instance, in PowerPoint presentations and slide shows. More precisely, consider two grayscale images of the same size, represented by the matrices and. For each scalar real number in the interval , define the matrix. Notice that , and, for each between and , the elements of the matrix are between the elements of the matrices and. Therefore, when varies from to , the matrix varies from to. For the case of color images, the transformation above must be applied to the matrices , and that compose each image.

Multiplication of matrices also has applications in digital image processing. Although our next example will be more elaborate with a rationale based on more advanced mathematical techniques usually only studied in Linear Algebra university courses , we believe, still, that it will be of interest to the reader, since this will have the opportunity to enjoy an amazing application derived from the ability to decompose a matrix as a product of matrices with special structures.

The omitted details may be found in the references [Lay, ] and [Poole, ]. Consider, therefore, a singular value decomposition SVD , that consists of writing a matrix as the product of three matrices:. Here is an example of an SVD decomposition:. Moreover, algorithms exist that allow us to calculate such decompositions using a computer. The key point of our example is to observe that if are the columns of the matrix and are the columns of the matrix then.

Why is that? Suppose that , a grayscale image of size , must be transmitted from a satellite to a laboratory on Earth. In principle, the satellite would have to send million numbers one for each pixel. Upon receiving these data, the laboratory on Earth calculates the matrix that will give an approximation of the original image. From the SVD decomposition of the corresponding matrix of this image, we can calculate the matrices for and. These matrices generate approximations to the original image, as illustrated in the following figures. Notice that the original image corresponds to the case. It is quite impressive, is it not? Digital image processing has many applications, such as remote sensing, data transmission, medicine, robotics, computer vision, film industry, etc.

In remote sensing, for example, images acquired by satellites are useful for tracking natural resources, geographical mapping, analysis of urban growth, and many other environmental applications. In the transmission of images, we have communications via fax, networks, the internet, and closed-circuit TV for monitoring and security. In medical applications, we have X-ray image processing, projection images of transaxial tomography, radiology, nuclear magnetic resonance NMR and ultrasonic scanning. Some methods of acquisition and transmission can generate noise in an image.

The median filter is an image processing technique used to remove them or reduce their effects: for each element of the matrix that represents the image, we observe its neighboring elements and, then, we arrange them in an ordered list. The median filter consists of choosing the central element of this list and replace the element at the center by this one. Figure image with noise and image with the median filter. There are many other techniques in image processing with different objectives. The following images illustrate examples of contrast adjustment, edge detection and threshold. Figure original picture, and the same picture with contrast adjustment, edge detection and threshold. Our objective with this text is to present a little known application of matrices for teachers and high school students: the processing of digital images.

It is important to notice that the mathematical tools related to this topic go far beyond matrices. The subject is vast, rich and modern. Unfortunately, the limit of few pages recommended for this article does not allow us to provide further details. As a starting point for the reader motivated to find out more, we recommend the books [Gonzalez and Woods, ] and [Gomes and Velho, ].

On this website or at its mirror , there is a series of interactive applets that allows you to explore the relations between matrices and digital images presented in this text. You will also find a DOC file with suggestions of exercises to be worked in the classroom. Gomes, J. Image Processing for Computer Graphics and Vision. Springer-Verlag, Gonzalez, R. Digital Image Processing. Third Edition. Prentice Hall, Excellent, I really enjoyed this article. In epistemology , and often in common use, the prefix meta- is used to mean about its own category. For example, metadata are data about data who has produced them, when, what format the data are in and so on. In a database, metadata are also data about data stored in a data dictionary and describe information data about database tables such as the table name, table owner, details about columns, — essentially describing the table.

Also, metamemory in psychology means an individual's knowledge about whether or not they would remember something if they concentrated on recalling it. The modern sense of "an X about X" has given rise to concepts like " meta-cognition " cognition about cognition , " meta-emotion " emotion about emotion , " meta-discussion " discussion about discussion , " meta-joke " joke about jokes , and " metaprogramming " writing programs that manipulate programs. In a rule-based system , a metarule is a rule governing the application of other rules. Any subject can be said to have a metatheory , a theoretical consideration of its properties, such as its foundations , methods , form and utility , on a higher level of abstraction.

In linguistics, a grammar is considered as being expressed in a metalanguage , language operating on a higher level to describe properties of the plain language and not itself. The earliest form of the word "meta" is the Mycenaean Greek me-ta , written in Linear B syllabic script. Its use in English is the result of back-formation from the word "metaphysics". In origin Metaphysics was just the title of one of the principal works of Aristotle ; it was so named by Andronicus of Rhodes because in the customary ordering of the works of Aristotle it was the book following Physics ; it thus meant nothing more than "[the book that comes] after [the book entitled] Physics ". However, even Latin writers misinterpreted this as entailing metaphysics constituted "the science of what is beyond the physical".

The use of the prefix was later extended to other contexts based on the understanding of metaphysics to mean "the science of what is beyond the physical". The Oxford English Dictionary cites uses of the meta- prefix as "beyond, about" such as meta-economics and meta-philosophy going back to However, these formations are parallel to the original "metaphysics" and "metaphysical", that is, as a prefix to general nouns fields of study or adjectives. Going by the OED citations, it began being used with specific nouns in connection with mathematical logic sometime before In David Hilbert proposed a research project in what was called " metamathematics. A notable early citation is Quine 's use of the word "metatheorem", [6] where meta- has the modern meaning of "an X about X".

Note earlier uses of "meta-economics" and even "metaphysics" do not have this doubled conceptual structure — they are about or beyond X but they do not themselves constitute an X.

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